#513036

order	512
nilpotency class	5
isoclinism	isoclinic class of #512941
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33720	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

400 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(10,17)(11,23)(12,13)(14,28)(15,34)(19,37)(20,38)(21,48)(22,47)(24,25)(26,57)(27,51)(29,59)(30,33)(31,32)(35,58)(36,41)(39,46)(42,61)(43,66)(44,63)(45,69)(49,52)(50,55)(53,70)(54,56)(60,64)(62,71)(65,72)(67,68);(2,5)(6,8)(9,10,13,20,14,21,31,47,34,46,59,69,58,63,30,37,16,19,33,44,35,45,29,39,15,22,32,48,28,38,12,17)(11,18,23,41,25,42,49,67,55,66,57,71,70,72,54,60,27,40,51,64,56,65,53,62,26,43,50,68,52,61,24,36);(1,3,4,7)(2,6,5,8)(9,11,12,24,28,52,32,50,15,26,29,53,35,56,33,51,16,27,30,54,58,70,59,57,34,55,31,49,14,25,13,23)(10,18,17,36,38,61,48,68,22,43,39,62,45,65,44,64,19,40,37,60,63,72,69,71,46,66,47,67,21,42,20,41);(10,19)(17,37)(18,40)(20,44)(21,45)(22,46)(36,60)(38,63)(39,47)(41,64)(42,65)(43,66)(48,69)(61,72)(62,67)(68,71)

Polycyclic presentation:

2^9|b^2=f,c^2=e,e^2=gh,f^2=g,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,h^a=hi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232017, #72322401, #72322434
maximal normal subgroup of	#72232017, #72322401, #72322434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420232);